Properties

Limits and colimits and closure

If DD has limits or colimits of a certain shape, then so does [C,D][C,D] and they are computed pointwise. (However, if DD is not complete, then other limits in [C,D][C,D] can exist “by accident” without being pointwise.)

κ\kappa-accessible functors from a κ\kappa-accessible category to any locally λ\lambda-presentable category form a locally λ\lambda-presentable category.

Cocontinuous functors between locally presentable categories form a locally presentable category. More precisely, if CC and DD are locally κ\kappa-presentable, then so is [C,D][C,D].

Continuous accessible functors between locally presentable categories form the opposite of a locally presentable category. More precisely, if CC and DD are locally κ\kappa-presentable, then so is [C,D]rmop[C,D]^{\rm op}.

Indeed, the point is this: given a κ\kappa-accessible category𝒞≃Indκ(𝒜)\mathcal{C} \simeq Ind^\kappa (\mathcal{A}) (𝒜\mathcal{A} essentially small), the category of κ\kappa-accessible functors 𝒞→𝒟\mathcal{C} \to \mathcal{D} (for arbitrary 𝒟\mathcal{D}; here by “κ\kappa-accessible” we mean simply “preserves κ\kappa-filtered colimits”) is naturally equivalent to the category of all 𝒜→𝒟\mathcal{A} \to \mathcal{D}. It should be well known that:

If 𝒟\mathcal{D} is accessible, then so is [𝒜,𝒟][\mathcal{A}, \mathcal{D}].

If 𝒟\mathcal{D} is locally λ\lambda-presentable, then so is [𝒜,𝒟][\mathcal{A}, \mathcal{D}].

Colimit-preserving functors out of a locally κ\kappa-presentable category are κ\kappa-accessible.

A right adjoint between locally κ\kappa-presentable categories is κ\kappa-accessible if and only if its left adjoint is strongly κ\kappa-accessible (i.e. preserves κ\kappa-presentable objects as well as κ\kappa-filtered colimits); and every limit-preserving accessible functor between locally presentable categories is a right adjoint.

Statements 1 and 2 are proved in [Adamek and Rosický, Locally presentable and accessible categories], statement 3 is obvious, and statement 4 is a straightforward exercise. Thus the claims follow.

In general, accessible functors between accessible categories do not form an accessible category due to size issues. The best one can hope for is a class-accessible category. Let 𝒞\mathcal{C} be an accessible category that is not essentially small. Consider the category 𝒜\mathcal{A} of all accessible functors 𝒞→Set\mathcal{C} \to \mathbf{Set}. This is the same as the smallest full replete subcategory of [𝒞,Set][\mathcal{C}, \mathbf{Set}] containing all representable functors and closed under small colimits. In particular, 𝒜\mathcal{A} is accessible if and only if 𝒜\mathcal{A} locally presentable. We claim 𝒜\mathcal{A} is not accessible.

Indeed, suppose 𝒜\mathcal{A} has a small generating family, say 𝒢\mathcal{G}. Then for some regular cardinal κ\kappa, every member of 𝒢\mathcal{G} is κ\kappa-accessible. So consider 𝒞(X,−)\mathcal{C} (X, -) for some object XX that is notκ\kappa-presentable. (Such an XX exists because 𝒞\mathcal{C} is not essentially small.) Since 𝒢\mathcal{G} generates, there is a small diagram of κ\kappa-accessible functors whose colimit is 𝒞(X,−)\mathcal{C} (X, -). But then 𝒞(X,−)\mathcal{C} (X, -) is a retract of a κ\kappa-accessible functor and hence κ\kappa-accessible: a contradiction. That said, 𝒜\mathcal{A} is a class-locally presentable category.